100年 - [非官方正解]100 國立臺灣大學轉學生招生考試試題_(商、管、農、經學院聯招) ：微積分(C)#113228

1.已知3x²+2xy+y²=1， dy/dx 等於下列哪一個選項? (A) (B)(C) (D)(E) .

2.下述五個選項中,那個選項中的x取值,使得級數收斂?  (A)除了x=0之外(B)x=3(C)-3≤x≤3(D)|x|＞3 (E)對所有的x,該級數都發散。

3.已知df(x)/dx=g(x)且h(x)=x³,則等於下列哪一個選項? (A)g(x³)(B)3x²g(x)(C)g'(x)(D)3x²g(x³)(E)x³g(x³)。

4.就下述之極限而言,哪一個選項正確?(A)極限為1(B)極限為2(C)極限為0(D)極限為-1(E)極限不存在。

5.若對所有的實數x,F'(x)皆為連續函數,則等於下列哪一個選項? (A) 0 (B) F(0) (C) F(3) (D) F'(0) (E) F'(3)ㆍ

7. Let g be the function given by g(x) =dt. Which of the following statements about g must be true?
I. g is increasing on the interval (1,2).
Ⅱ.g is increasing on the interval (2,3).
III. g(3) ＞0 (A) I only (B) II only (C) II only (D) II and IIl only (E) I, I, and III。

9.當x=f(x,y)且x=rcosθ及y=rsinθ。請問下列哪些選項是正確的?

(A) (B)

(C)

(D)

(E)

10.令l代表下列積分請問下列哪些選項是正確的? (A) I = (B)I= (C)I=∫∫Ω(x²+y²)dA,其中Ω為以(1,0)為圓心,半徑為1的圓盤 (D) I =(E) I ≤4ㅠ 

11. Let A be the region bounded above by the graph of y = sin(πx) and below by the graph of y = x³ - 4x. (A)(2.1,0.8) falls into the region A. (B) The area of A can be represented bydx. (C) The area of A is equal to 4. (D)The region A is the base of a solid. For this solid, each cross section per- pendicular to the a-axis is a square.The volume of this solid is equal to dx. (E)The region A models the surface of a small pond. At all points in A tance :i from the y-axis, the depth of the water is given by h(x) = 3-x. The yolume of water in the pond is equal todx.

(b) Find limr→∞ I(r).(5分)